Best way to add height with textures?
The reason why your tiles are becoming so rounded off, is because you have created the displacement map from a picture where there is a lot of reflections coming from one of the sides of each tile. Blender sees this as a height difference and this is your result. You'll either have to try getting those reflections out of the map (not recommended, because it's pretty difficult), or you'll have to raise the tiles manually. Also, displacement maps need a ton of faces. More than you would ever need to just raise flat tiles.

Good luck!